FOM: Toposy-turvey

Colin Mclarty cxm7 at
Sat Jan 17 15:18:46 EST 1998

Reply to message from sf at Csli.Stanford.EDU of Fri, 16 Jan
>I didn't think I would get drawn again into a discussion of category
>theory vs. set theory as a foundation of mathematics, but Aczel's FOM
>drugs are at work.  First, I appreciate Peter's effort to clarify what the
>conflict is about.  But I think he has left out something essential, and
>that was what I emphasized in an earlier related posting of 19 Nov 23:46.
>Namely, the notion of topos is a relatively sophisticated mathematical
>notion which assumes understanding of the notion of category and that in
>turn assumes understanding of notions of collection and function.

	To me, understanding the notion of category is the best way
known today of understanding the notions of "collection and function".

	To be precise: Pace Feferman's theory, the first thing you
need to know about collections and functions is that a pair of functions
may have a composite, and this composition is associative when it is 
defined. And now I have completely stated the category axioms. This 
seems to me not very sophisticated mathematics. The topos axioms are
more sophisticated, but no more so than ZF or Feferman's theory of
collections and functions.

>  I do
>not mean these in the sense identified in the cumulative hierarchy or any
>supposed standard model for that, but rather in the informal everyday
>mathematical sense.  Thus there is both a logical and psychological
>priority for the latter notions to the former.  'Logical' because what a
>topos is requires a definition in order to work with it and prove theorems
>about it, and this definition ultimately returns to the notions of
>collection (class, set, or whatever word you prefer) and function
>(or operation).  

	Actually, when you pose a topos as a foundation you do not
use a *definition* of a topos, you use *axioms* on objects and arrows.

	This is exactly the same point as saying that Feferman's
foundation begins not by *defining* a universe of collections and
operations, but by giving *axioms* on collections and functions.

>'Psychological' because you can't understand what a topos
>is unless you have some understanding of those notions.  Just writing down
>the "axioms" for a topos does not provide that understanding.  If that is
>granted, and I can't continue further in this discussion if it is not,
>then what work in f.o.m. has to accomplish is a clarification of the
>notions of collection (or class) and function (or operation).

	I am not sure which point Feferman needs granted before he
can go on. I certainly grant that writing down axioms does not by
itself provide understanding. I even grant that you need informal
understanding of a lot of things: arithmetic, functions, explicit
deductive reasoning, and more, before any kind of foundational axioms
will make sense to you. I do not believe that discrete structureless
collections stand out among mathematical objects as THE ONES you have
to know about to understand a foundation. 

	I wonder if that much disagreement really does itself preclude
meaningful discussion. It might but I hope not.


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